Solve (sqrt{2}-sqrt{3}+sqrt{5})(sqrt{2}+sqrt{3}-sqrt{5})

$$( \sqrt { 2 } - \sqrt { 3 } + \sqrt { 5 } ) ( \sqrt { 2 } + \sqrt { 3 } - \sqrt { 5 } )$$

Evaluate

$$2\left(\sqrt{15}-3\right)\approx 1.745966692$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{2}-\sqrt{3}+\sqrt{5} by each term of \sqrt{2}+\sqrt{3}-\sqrt{5}.

$$\left(\sqrt{2}\right)^{2}+\sqrt{2}\sqrt{3}-\sqrt{2}\sqrt{5}-\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2+\sqrt{2}\sqrt{3}-\sqrt{2}\sqrt{5}-\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2+\sqrt{6}-\sqrt{2}\sqrt{5}-\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$2+\sqrt{6}-\sqrt{10}-\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$2+\sqrt{6}-\sqrt{10}-\sqrt{6}-\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

Combine \sqrt{6} and -\sqrt{6} to get 0.

$$2-\sqrt{10}-\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2-\sqrt{10}-3+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

Subtract 3 from 2 to get -1.

$$-1-\sqrt{10}+\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$-1-\sqrt{10}+\sqrt{15}+\sqrt{5}\sqrt{2}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$-1-\sqrt{10}+\sqrt{15}+\sqrt{10}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

Combine -\sqrt{10} and \sqrt{10} to get 0.

$$-1+\sqrt{15}+\sqrt{5}\sqrt{3}-\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$-1+\sqrt{15}+\sqrt{15}-\left(\sqrt{5}\right)^{2}$$

Combine \sqrt{15} and \sqrt{15} to get 2\sqrt{15}.

$$-1+2\sqrt{15}-\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$-1+2\sqrt{15}-5$$

Subtract 5 from -1 to get -6.

$$-6+2\sqrt{15}$$